Internal semidirect product

This article describes a product notion for groups. See other related product notions for groups.

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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Definition

Definition with symbols

A group $G$ is termed an internal semidirect product of subgroups $N$ and $H$ if the following hold:

• $N$ is a normal subgroup of $G$
• $N$ and $H$ are permutable complements

Note here that $H$ acts as automorphisms on $N$ by the conjugation action.

Equivalence with external semidirect product

Further information: Equivalence of internal and external semidirect product

Suppose $G$ is an internal semidirect product with normal subgroup $N$ and $H$ as the other subgroup. If we start out with $N$ and $H$ as abstract groups, and with the action of $H$ on $N$ (abstractly) which comes from the conjugation in $G$, then the external semidirect product formed from these is isomorphic to $G$.

Terminology

• A subgroup which occurs as the normal subgroup for an internal semidirect product is termed a complemented normal subgroup, sometimes also called split normal subgroup.
• A subgroup which occurs as the permutable complement to a normal subgroup, is termed a retract. This is because there is a retraction from the whole group, to this subgroup, whose kernel is the normal subgroup.

Examples

Trivial examples

• Every group is the internal semidirect product of itself and the trivial subgroup. In fact, it is an internal direct product of itself and the trivial subgroup.
• Given two groups $G$ and $H$, $G\times H$ is the internal semidirect product of $G\times \{e\}$ and $\{e\}\times H$. In fact, it is the internal direct product.

Simple examples

• The symmetric group on any finite set of size at least two is the internal semidirect product of the alternating group and the two-element subgroup generated by any transposition. For instance, the symmetric group of degree three is the internal semidirect product of the subgroups $\{(),(1,2,3),(1,3,2)\}$ and $\{(),(1,2)\}$.
• The symmetric group of degree four is an internal semidirect product of the normal subgroup $\{(),(1,2)(3,4),(1,3)(2,4),(1,4)(2,3)\}$ and a six-element subgroup (isomorphic to symmetric group of degree three) comprising the permutations that fix $4$.
• The dihedral group of degree $n$ and order $2n$ is the internal semidirect product of a cyclic subgroup of order $n$ (the rotations) and a cyclic subgroup of order $2$ (generated by a reflection).

Non-examples

• The cyclic group of order four has a normal subgroup of order two, but this normal subgroup has no complement.

Relation with other properties

Stronger product notions

• Internal regular product
• Internal direct product

Weaker product notions

• Exact factorization
• Group extension

Related subgroup properties

• Complemented normal subgroup is a normal subgroup having a permutable complement, and hence, part of a semidirect product.
• Retract is a subgroup having a normal complement, and hence, part of a semidirect product.

Related group properties

• Splitting-simple group is a group that cannot be expressed as an internal semidirect product of nontrivial subgroups.

Facts

• Complement to normal subgroup is isomorphic to quotient: In particular, given the normal subgroup for a semidirect product, we know the isomorphism type of the complement.
• Complements to normal subgroup need not be automorphic: Given a normal subgroup $N$ of a group $G$ with two complements $H_1$ and $H_2$, it is not necessary that there exist an isomorphism of $G$ sending $H_1$ to $H_2$. It is also not necessary that the actions of $H_1$ and $H_2$ on $N$ are the same.
• Complements to abelian normal subgroup are automorphic
• Complements to normal Hall subgroup are conjugate: This is the conjugacy part of the result commonly known as the Schur-Zassenhaus theorem.